@c Copyright (C) 1996-2022 The Octave Project Developers
@c
@c This file is part of Octave.
@c
@c Octave is free software: you can redistribute it and/or modify it
@c under the terms of the GNU General Public License as published by
@c the Free Software Foundation, either version 3 of the License, or
@c (at your option) any later version.
@c
@c Octave is distributed in the hope that it will be useful, but
@c WITHOUT ANY WARRANTY; without even the implied warranty of
@c MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
@c GNU General Public License for more details.
@c
@c You should have received a copy of the GNU General Public License
@c along with Octave; see the file COPYING.  If not, see
@c <https://www.gnu.org/licenses/>.

@node Arithmetic
@chapter Arithmetic

Unless otherwise noted, all of the functions described in this chapter
will work for real and complex scalar, vector, or matrix arguments.  Functions
described as @dfn{mapping functions} apply the given operation individually to
each element when given a matrix argument.  For example:

@example
@group
sin ([1, 2; 3, 4])
     @result{}  0.84147   0.90930
         0.14112  -0.75680
@end group
@end example

@menu
* Exponents and Logarithms::
* Complex Arithmetic::
* Trigonometry::
* Sums and Products::
* Utility Functions::
* Special Functions::
* Rational Approximations::
* Coordinate Transformations::
* Mathematical Constants::
@end menu

@node Exponents and Logarithms
@section Exponents and Logarithms

@DOCSTRING(exp)

@DOCSTRING(expm1)

@DOCSTRING(log)

@DOCSTRING(reallog)

@DOCSTRING(log1p)

@DOCSTRING(log10)

@DOCSTRING(log2)

@DOCSTRING(pow2)

@DOCSTRING(nextpow2)

@DOCSTRING(realpow)

@DOCSTRING(sqrt)

@DOCSTRING(realsqrt)

@DOCSTRING(cbrt)

@DOCSTRING(nthroot)

@node Complex Arithmetic
@section Complex Arithmetic

In the descriptions of the following functions,
@tex
$z$ is the complex number $x + iy$, where $i$ is defined as
$\sqrt{-1}$.
@end tex
@ifnottex
@var{z} is the complex number @var{x} + @var{i}@var{y}, where @var{i} is
defined as @code{sqrt (-1)}.
@end ifnottex

@DOCSTRING(abs)

@DOCSTRING(arg)

@DOCSTRING(conj)

@DOCSTRING(cplxpair)

@DOCSTRING(imag)

@DOCSTRING(real)

@node Trigonometry
@section Trigonometry

Octave provides the following trigonometric functions where angles are
specified in radians.  To convert from degrees to radians multiply by
@tex
$\pi/180$
@end tex
@ifnottex
@code{pi/180}
@end ifnottex
or use the @code{deg2rad} function.  For example, @code{sin (30 * pi/180)}
returns the sine of 30 degrees.  As an alternative, Octave provides a number of
trigonometric functions which work directly on an argument specified in
degrees.  These functions are named after the base trigonometric function with
a @samp{d} suffix.  As an example, @code{sin} expects an angle in radians while
@code{sind} expects an angle in degrees.

Octave uses the C library trigonometric functions.  It is expected that these
functions are defined by the ISO/IEC 9899 Standard.  This Standard is available
at: @url{http://www.open-std.org/jtc1/sc22/wg14/www/docs/n1124.pdf}.
Section F.9.1 deals with the trigonometric functions.  The behavior of most of
the functions is relatively straightforward.  However, there are some
exceptions to the standard behavior.  Many of the exceptions involve the
behavior for -0.  The most complex case is atan2.  Octave exactly implements
the behavior given in the Standard.  Including
@tex
$atan2(\pm0, -0)$ returns $\pm \pi$.
@end tex
@ifnottex
@code{atan2(+- 0, 0)} returns @code{+- pi}.
@end ifnottex

It should be noted that @sc{matlab} uses different definitions which apparently
do not distinguish -0.

@DOCSTRING(deg2rad)
@DOCSTRING(rad2deg)

@DOCSTRING(sin)
@DOCSTRING(cos)
@DOCSTRING(tan)
@DOCSTRING(sec)
@DOCSTRING(csc)
@DOCSTRING(cot)

@DOCSTRING(asin)
@DOCSTRING(acos)
@DOCSTRING(atan)
@DOCSTRING(asec)
@DOCSTRING(acsc)
@DOCSTRING(acot)

@DOCSTRING(sinh)
@DOCSTRING(cosh)
@DOCSTRING(tanh)
@DOCSTRING(sech)
@DOCSTRING(csch)
@DOCSTRING(coth)

@DOCSTRING(asinh)
@DOCSTRING(acosh)
@DOCSTRING(atanh)
@DOCSTRING(asech)
@DOCSTRING(acsch)
@DOCSTRING(acoth)

@DOCSTRING(atan2)

Octave provides the following trigonometric functions where angles are
specified in degrees.  These functions produce true zeros at the appropriate
intervals rather than the small round-off error that occurs when using
radians.  For example:

@example
@group
cosd (90)
     @result{} 0
cos (pi/2)
     @result{} 6.1230e-17
@end group
@end example

@DOCSTRING(sind)
@DOCSTRING(cosd)
@DOCSTRING(tand)
@DOCSTRING(secd)
@DOCSTRING(cscd)
@DOCSTRING(cotd)

@DOCSTRING(asind)
@DOCSTRING(acosd)
@DOCSTRING(atand)
@DOCSTRING(atan2d)
@DOCSTRING(asecd)
@DOCSTRING(acscd)
@DOCSTRING(acotd)

Finally, there are two trigonometric functions that calculate special
arguments with increased accuracy.

@DOCSTRING(sinpi)
@DOCSTRING(cospi)

@node Sums and Products
@section Sums and Products

@DOCSTRING(sum)

@DOCSTRING(prod)

@DOCSTRING(cumsum)

@DOCSTRING(cumprod)

@DOCSTRING(sumsq)

@node Utility Functions
@section Utility Functions

@DOCSTRING(ceil)

@DOCSTRING(fix)

@DOCSTRING(floor)

@DOCSTRING(round)

@DOCSTRING(roundb)

@DOCSTRING(max)

@DOCSTRING(min)

@DOCSTRING(cummax)

@DOCSTRING(cummin)

@DOCSTRING(hypot)

@DOCSTRING(gradient)

@DOCSTRING(dot)

@DOCSTRING(cross)

@DOCSTRING(divergence)

@DOCSTRING(curl)

@DOCSTRING(del2)

@DOCSTRING(factorial)

@DOCSTRING(factor)

@DOCSTRING(gcd)

@DOCSTRING(lcm)

@DOCSTRING(rem)

@DOCSTRING(mod)

@DOCSTRING(primes)

@DOCSTRING(list_primes)

@DOCSTRING(sign)

@DOCSTRING(signbit)

@node Special Functions
@section Special Functions

@DOCSTRING(airy)

@DOCSTRING(besselj)

@DOCSTRING(bessely)

@DOCSTRING(besseli)

@DOCSTRING(besselk)

@DOCSTRING(besselh)

@DOCSTRING(beta)

@DOCSTRING(betainc)

@DOCSTRING(betaincinv)

@DOCSTRING(betaln)

@DOCSTRING(bincoeff)

@DOCSTRING(commutation_matrix)

@DOCSTRING(cosint)

@DOCSTRING(duplication_matrix)

@DOCSTRING(dawson)

@DOCSTRING(ellipj)

@DOCSTRING(ellipke)

@DOCSTRING(erf)

@DOCSTRING(erfc)

@DOCSTRING(erfcx)

@DOCSTRING(erfi)

@DOCSTRING(erfinv)

@DOCSTRING(erfcinv)

@DOCSTRING(expint)

@DOCSTRING(gamma)

@DOCSTRING(gammainc)

@DOCSTRING(gammaincinv)

@DOCSTRING(legendre)

@anchor{XREFgammaln}
@DOCSTRING(lgamma)

@DOCSTRING(psi)

@DOCSTRING(sinint)

@node Rational Approximations
@section Rational Approximations

@DOCSTRING(rat)

@DOCSTRING(rats)

@node Coordinate Transformations
@section Coordinate Transformations

@DOCSTRING(cart2pol)

@DOCSTRING(pol2cart)

@DOCSTRING(cart2sph)

@DOCSTRING(sph2cart)

@node Mathematical Constants
@section Mathematical Constants

@DOCSTRING(e)

@DOCSTRING(pi)

@DOCSTRING(I)

@DOCSTRING(Inf)

@DOCSTRING(NaN)

@DOCSTRING(eps)

@DOCSTRING(realmax)

@DOCSTRING(realmin)
